A382493 a(n) is the 2-adic valuation of the least number that has exactly n 3-smooth divisors.

0, 1, 2, 1, 4, 2, 6, 3, 2, 4, 10, 3, 12, 6, 4, 3, 16, 5, 18, 4, 6, 10, 22, 5, 4, 12, 8, 6, 28, 5, 30, 7, 10, 16, 6, 8, 36, 18, 12, 7, 40, 6, 42, 10, 8, 22, 46, 7, 6, 9, 16, 12, 52, 8, 10, 7, 18, 28, 58, 9, 60, 30, 8, 7, 12, 10, 66, 16, 22, 9, 70, 11, 72, 36, 14

Offset: 1

Author

Amiram Eldar, Mar 29 2025

Keywords

• nonn
• easy

Comments

a(n) = d-1, where d is the divisor of n that minimizes 2^(x-1)*3^(n/x-1).

The 3-adic valuation of A382492(n) is A007949(A382492(n)) = n/(a(n)+1) - 1.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007814, A007949, A037143, A099311, A382492.

Programs

Mathematica

a[n_] := IntegerExponent[Min[Table[2^(d-1)*3^(n/d-1), {d, Divisors[n]}]], 2]; Array[a, 100]

PARI

a(n) = valuation(vecmin(apply(d -> 2^(d-1)*3^(n/d-1), divisors(n))), 2);

Formula

a(n) = A007814(A382492(n)).

a(n) >= A033677(n) - 1 (inequality occurs at n = 36, 72, 120, 144, 180, 210, 240, ...).

a(p) = p-1 for prime p.

a(n) = A099311(n) if n is in A037143.